Symmetry constraints on many-body localization
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Symmetry constraints on many-body localization
Andrew C. Potter and Romain Vasseur , Department of Physics, University of California, Berkeley, CA 94720, USA and Materials Science Division, Lawrence Berkeley National Laboratories, Berkeley, CA 94720
We derive general constraints on the existence of many-body localized (MBL) phases in thepresence of global symmetries, and show that MBL is not possible with symmetry groups thatprotect multiplets (e.g. all non-Abelian symmetry groups). Based on simple representation theoreticconsiderations, we derive general Mermin-Wagner-type principles governing the possible alternativefates of non-equilibrium dynamics in isolated, strongly disordered quantum systems. Our resultsrule out the existence of MBL symmetry protected topological phases with non-Abelian symmetrygroups, as well as time-reversal symmetry protected electronic topological insulators, and in factall fermion topological insulators and superconductors in the 10-fold way classification. Moreover,extending our arguments to systems with intrinsic topological order, we rule out MBL phases withnon-Abelian anyons as well as certain classes of symmetry enriched topological orders.
The concept of symmetry plays a crucial role in ourunderstanding of phases of matter. The interplay ofsymmetry and dimensionality leads to very general con-straints on possible types of symmetry breaking phasesand phase transitions, such as the Peierls or Mermin-Wagner theorems. Going beyond Landau’s theory ofphases and phases transitions in terms of spontaneoussymmetry breaking, it was recently understood that sym-metries can protect topological distinctions among short-range entangled phases of matter – leading to the con-cept of symmetry protected topological (SPT) phases exemplified by the celebrated electronic topological insu-lators (TI) – and that symmetry can also enrich thepossibilities of quantum phases with long-range entangle-ment and intrinsic topological order .Whereas traditionally, the existence of phases andphase transitions is considered within the framework ofequilibrium statistical mechanics, a sharp notion of quan-tum phases can be extended to a certain class of far-from-equilibrium quantum systems that fail to self-thermalizein isolation. Highly excited eigenstates in such many-body localized (MBL) systems have properties akin toquantum groundstates , leading to the prospect of quan-tum coherent phenomena and universal dynamics ,and symmetry-breaking, topological or SPT quantum or-ders at high energy density .Given the fundamental role of symmetry in our under-standing of equilibrium phases of matter, it is natural toexpect very general symmetry principles to play a crucialrole in MBL systems. While certain examples of sym-metry based constraints on localization have been iden-tified , a full and systematic understanding remainsto be obtained. In this paper, we argue that the localconserved quantities that define MBL systems transformindependently under the global symmetry, leading to anextensive number of local degeneracies in the presenceof non-Abelian symmetries. This local action leads tovery general constraints that dictate the fate of the ex-cited state dynamics of strongly disordered systems onsymmetry grounds alone. Namely, we show that sym-metry preserving MBL phases are not possible with non-Abelian symmetries, either discrete or continuous. This eliminates out the possibility of SPT and symmetry en-riched topological (SET) order protected by non-Abeliansymmetries, and for instance means that it is impossibleto localize spin- electrons with time-reversal symme-try, ruling out the realization of electronic TIs in MBLsettings. Moreover, based on the representation theoryof the symmetry group, we derive very general Mermin-Wagner type principles governing the stability and pos-sible fates of strongly disordered systems with symme-try. For example, systems with continuous non-Abeliansymmetries inevitably thermalize even at strong disor-der, whereas systems with discrete non-Abelian symme-tries may yield non-ergodic phases that are either well lo-calized but with spontaneous symmetry breaking or aredelocalized and quantum-critical. Our results also con-straint the emergence of non-equilibrium quantum phaseswith intrinsic topological order and anyonic excitations,with the notion of anyonic fusion algebras replacing thegroup representation theory. For example, we rule outthe possibility of MBL with non-Abelian anyonic excita-tions. I. MANY-BODY LOCALIZATION ANDGLOBAL SYMMETRIESA. Many-body localization and local conservedquantities
We begin by fixing a formal definition of MBL interms of a complete set of quasi-local conservationlaws , which has been widely adopted, and fromwhich all of the known phenomenology of MBL systemssuch as area-law entanglement , absence of transport,slow dephasing and entanglement growth directlyfollow. Specifically, we define MBL in terms of the ex-istence of a complete set of conserved quantities, { n α } ( α = 1 . . . N ), each of which takes values from a set { , . . . , d α } , with the associated quasi-local conservedprojectors called l-bits Π n α α = d β X n β = α =1 | n . . . n α . . . n N i h n . . . n α . . . n N | , (1)that are exponentially well-localized within a localizationlength ξ of position r α and that commute with the Hamil-tonian [Π n α α , H ] = 0. We further assume that the quasi-local conserved quantities are related to local operators(with finite support) by dressing them with a quasi-localunitary transformation, which produces their exponentialtails. This structure is obtained in all known examplesof MBL, and in fact can be taken as the definition of thel-bits being quasi-local.Note that the number of conserved quantities N canin general be different from the number of physical sites L . We will restrict our attention the case where all ofthe energy eigenstates (or quasi-energy eigenstates forFloquet systems ) are MBL, and neglect the theoret-ically more delicate case of a partially localized spectrumwith a many-body mobility edge . In this case, themany-body eigenstates can be uniquely labeled by prod-uct states with definite values of the conserved quanti-ties: H |{ n α }i = E ( { n α } ) |{ n α }i , where E is a quasi-local function of n α . In the following, we will investigatewhether MBL can be compatible with additional exactdegeneracies due to symmetry. B. Local symmetry action
Having defined MBL we now derive some general con-straints on many-body localization in the presence ofsymmetry. To begin, we start by showing that symmetryacts locally on the l-bits.Consider a lattice of sites i containing quantum degreesof freedom that transform under a (possibly reducible)representation, V , of a symmetry group G – for exam-ple a chain of spins- ( V ), with spin-rotation symmetry G = SU (2). The Hilbert space of this system decomposesinto a tensor product of on-site Hilbert spaces, H = V ⊗ L .In order that different symmetry operations have non-trivial action on the physical degrees of freedom, we willdemand that V is a faithful representation of the sym-metry G , and that we cannot merely group degrees offreedom into larger clumps that transform under a sim-pler symmetry G ′ . The former condition implies that allirreducible representations (irreps) of G are contained inthe tensor product V ⊗ n for sufficiently large n .We define a symmetry preserving MBL phase as onein which the local conserved quantities labelling is con-sistent with the symmetry [Π n α α , Q i g i ] = 0, where g i ∈ V is the representation of the symmetry generator g ∈ G on the site i . This is equivalent to having each setof conserved quantities { n α } label a multiplet of states V n ,n ,...,n N that form a representation of G . To con-struct the local action of the symmetry, let us proceed as follows. Let us assume that at least one of the eigen-states labelled by a given set { n α } is non-degenerate andtransforms trivially under the symmetry, so that the cor-responding representation V n ,n ,...,n N = V is the trivialrepresentation (or singlet) with dimension 1. We thencreate a local excitation by changing the label n α to n α for a given l-bit α : the resulting eigenstate(s) then trans-form in a different representation V n α ≡ V n ,...,n α ,...,n N which will generically be irreducible (otherwise, onemay add generic local perturbations to reduce it). Be-cause the change n α → n α is local, all the eigenstates (cid:12)(cid:12) n , . . . , n α , . . . , n N ; p (cid:11) with p = 1 , . . . , dim V n α in thisrepresentation and the singlet eigenstate (cid:12)(cid:12) { n α } (cid:11) shoulddiffer only locally around r α . Let us now repeat the pro-cess to excite a different l-bit, β , and let V n α n β be therepresentation corresponding to the configuration wherewe changed the labels n α → n α and n β → n β on twodifferent locations α and β .We now show that the symmetry action factorizes onthese two excitations. Since by their definition any quasi-local set of l-bits can obtained by “dressing” strictly localoperators by a finite depth (quasilocal) unitary transfor-mation, it suffices to consider the case of strictly locall-bits ( i.e. with support only on a finite number of sites).For this case, it is clear that if r α and r β are sufficientlyfar apart so that the corresponding supports do not over-lap, then the action of the symmetry factorizes on the twol-bits α and β : V n α n β = V n α ⊗ V n β . Repeating the ar-gument for an extensive number of l-bits α , α , . . . , α p with p ∼ O ( N ) ∼ O ( L ) sufficiently far apart so that theirsupport do not overlap, we find that the symmetry actionfactorizes on the local l-bits V n α ...n αp = V n α ⊗· · ·⊗V n αp . C. Examples of local symmetry action
This local factorization of the symmetry on the l-bitsis particularly obvious for models of MBL paramagnets,such as the Ising paramagnet in arbitrary dimension with G = Z H = − L X i =1 h i σ xi + . . . (2)where the dots represent small (but arbitrary) symmetry-preserving perturbations. The eigenstates of Eq. 2 arerelated to product states of definite σ xi = ± U , such thatthe local conserved quantities of this MBL systems arethe dressed projectors Π n =0 , α i = U † ± σ xi U . In this case,the local action of the symmetry is simply given byˆ g α i = U † σ xi U, (3)with [ˆ g α i , H ] = 0, since ˆ g α i commutes with all the con-served quantities Π nα j , readily verifying that the globalsymmetry [ Q i g i , H ] = 0 with g i = σ xi is promoted to alocal symmetry [ˆ g α i , H ] = 0 for the MBL system.This construction can be readily generalized tothe generic MBL paramagnet Hamiltonian H para = − P i P n h ni P ni + . . . where the P ni ’s are projection oper-ators onto the different irreps (“channels”) in the decom-position V = ⊕ n V n ( P n P ni = 1) of the on-site represen-tation V of G , and the dots represent generic weak per-turbations. As in the Ising example, the local conservedquantities of this MBL system are the dressed projec-tors Π nα i = U † P ni U where U is a finite depth (quasilocal)unitary transformation, and the local action of the sym-metry is simply given by ˆ g α i = U † g i U , where g i in therepresentation of the group element g on site i .A less straightforward example are SPT phases, whichcannot be continuously connected to a trivial paramag-net while preserving symmetry. However, the SPT eigen-states are non-continuously deformable, via a finite-depthunitary transformation U SPT that preserves the symme-try everywhere in the bulk, to a trivial paramagnet (seee.g. , and Appendix A for a specific example). We canthen utilize the construction for paramagnets to iden-tify, the local action of symmetry as being generated byˆ g α,i = U † U † SPT g i U SPT U , where U is the quasi-local uni-tary that dresses the l-bits. These generators form an or-dinary local representation of symmetry in the bulk, butact non-trivially (e.g. projectively in 1D) at the edges ofthe system. II. MBL AND NON-ABELIAN SYMMETRY
The local factorization of the symmetry on the l-bitshas important consequences when G is non-Abelian, forwhich some irreps are necessarily multidimensional. In-tuitively, this signals an obstacle to localization, sincea generic MBL state will contain many of these multidi-mensional excitations, each with local degrees of freedomthat cost no energy to excite, and can therefore freelyinter-resonate with each other leading to a breakdown oflocalization. This rules out the existence of MBL param-agnets with Potts (permutation group G = S n ) or non-chiral clock (Dihedral group G = D n = Z n ⋊ Z ) symme-try for instance. We emphasize that while our argumentrelied on the local integrability picture of MBL systems,we expect the main idea to be fairly general so that itwould also rule out tentative MBL phases without anl-bit description (see discussion below).More formally, if we were to have an MBL system withnon-Abelian symmetry G , then the local conserved quan-tities would transform as irreps of the symmetry group G so that each l-bit, n α , labels an irrep V n α of G . TheHilbert space therefore has a symmetry preserving ten-sor structure in the l-bit space H = ⊗ α V α , where therepresentation V α is reducible and can be decomposed as V α = ⊕ d α n α =1 V n α . Since the physical degrees of freedomtransform in a faithful representation V of the symme-try, at least a finite density of the V α ’s should be faithfulrepresentations of G as well. If G is non-Abelian, thisimmediately implies that some irreps V n α should have dimension larger than 1 so that the quantum numbers n α must be supplemented with an additional number p α = 1 , . . . , D n α = dim V n α to label uniquely an eigen-state. This finite density of local multidimensional irrepsleads to an exponential degeneracy of the eigenstates of H since the energy cannot depend on the extra labels p α .In other words, the global symmetry G is promoted to alocal symmetry[ˆ g n α , Π n β β ] = [ˆ g n α , H ] = 0 , (4)because of the many-body localized structure of theeigenstates, with ˆ g n α being the representation of g ∈ G in V n α , acting locally around position r α . This extendedlocal symmetry leads to local degeneracies if there aremultidimensional irreps, leading in turn to a massiveexponential-in-system-size degeneracy of all eigenstates.Such degenerate eigenstates are inherently unstable, evento infinitesimally small, perturbations. However, thecrucial point is that there is no local and symmetry-preserving way to resolve this degeneracy. Hence, ei-ther the symmetry or the localization must break down.Which of these fates may occur depends on the groupstructure, and below, we will identify some simple gov-erning principles based on the number and dimensions ofthe irreps of the symmetry group.Note that whereas the above discussion assumed a fullfactorization of the symmetry on the l-bits for simplicity(so that the Hilbert space factorizes as H = ⊗ α V α ), theexistence of exponentially-degenerate eigenstates only re-quires a partial factorization of the symmetry for a gen-eral excitation involving an extensive number of l-bitsfar-enough apart V n α ...n αp = V n α ⊗ · · · ⊗ V n αp , whichwe showed in Sec. I B. III. GENERAL SYMMETRY PRINCIPLES
Above, we have shown that non-Abelian symmetriesare not consistent with MBL phases, and we now seeksome general insight into the possible fates of an isolatednon-equilibrium system with non-Abelian symmetries. Ifdisorder is too weak, we expect that the putative localdegenerate excitations will strongly overlap and inter-resonate, driving the system into a thermalizing phase.Hence, we will subsequently focus on the regime of strongdisorder, considering various classes of non-Abelian sym-metry groups in turn.When the non-Abelian group G has irreps of boundeddimension, with either infinitely many irreps (as for thegroup G = U (1) ⋊ Z where all irreps have dimension ≤
2) or a finite number of them (as for any finite non-Abelian symmetry), there are a few options to lift thedegeneracies at strong disorder. One possible outcomeis that the system forms an MBL state in which sym-metry is spontaneously broken down to an Abelian sub-group by choosing a particular set of the numbers p α .This spontaneous symmetry breaking (SSB) scenario waspreviously demonstrated for the particular example of G < ∞ ∞|V| = 1 MBL ✓ Ex: Z n MBL ✓ Ex: U (1)1 < |V| < ∞ MBL ✗ Ex: Z n ⋊ Z MBL ✗ Ex: U (1) ⋊ Z → MBL+SSB (or QCG?) → MBL+SSB |V| = ∞ N/A MBL ✗ Ex: SU (2) → Thermalization onlyTABLE I.
Symmetry constraints on MBL:
Possible phases of an isolated interacting system at strong disorder in termsof the representation theory of its symmetry group G . The relevant parameters are the number of irreps and the dimension ofthe largest irrep |V| = sup k { dim V k } . If G is Abelian ( |V| =1), then a many-body localized phase is possible at strong disorder.If on the other hand G is non-Abelian, a symmetry-preserving MBL phase is not allowed, giving rise to either thermalization,MBL with the symmetry spontaneously broken (SSB) to an Abelian subgroup, or to non-trivial quantum critical glasses (QCG)depending on the properties of the symmetry group (see text). a random XXZ spin chain, equivalent to fermions withparticle-hole symmetric disorder with symmetry group G = U (1) ⋊ Z using renormalization group techniquesand numerical simulations. Another possible option forfinite groups would be for the system to form a symmetrypreserving “quantum critical glass” (QCG) which is nei-ther thermal nor exponentially localized, and that cannotbe described in terms of independent conserved quanti-ties (examples of such phases have been uncovered inanalytically solvable random anyonic chains ). It wouldbe very interesting to find a concrete example of sucha QCG phase in a random spin chain with non-Abeliansymmetry. Note however that our argument also rulesout marginal or quantum critical MBL states where independent l-bits exist, just with algebraic ratherthan exponential tails (as for the critical random trans-verse field Ising chain for example ).If the non-Abelian symmetry group G is continuous,e.g. G = SU (2), then it will possess infinitely manyirreps with arbitrarily large dimension. E.g., for G = SU (2), irreps are labelled by spin size S and have di-mension 2 S + 1 where S can be arbitrarily large. Thenfollowing Ref. 36 in a large many-body system, one willencounter excitations with arbitrarily large local degen-eracy, D (large “spins”), whose quantum fluctuations aresuppressed as 1 /D , leading to effectively classical dynam-ics, and resulting in thermalization, even for arbitrarilystrong disorder . Note that it is furthermore not possi-ble to realize an MBL phase by spontaneously breakingthe continuous non-Abelian symmetry, as this would pro-duce a delocalized Goldstone mode that would act as abath , so that thermalization is the only possiblescenario.The only scenario that permits stable MBL phaseswith symmetry are Abelian groups, whose irreps all havedimension 1, avoiding the pitfalls of the above examples.These different scenarios are summarized in Tab. I. IV. CONSEQUENCES FORNON-EQUILIBRIUM TOPOLOGICAL PHASESA. Consequences for SPT order
The above-identified obstruction to MBL rules outthe possibility of localization stabilized SPT order (orFloquet SPT order ) with non-Abelian symmetrygroups such as the Haldane chains with continuous SO (3)symmetry – as these phases require both symmetry andMBL to occur at high energy density. This further con-strains the many-body localizability of SPT phases .We remark that these results apply also to anti-unitarysymmetries such as time reversal symmetry (TRS). Thenotion of local action of TRS is in general somewhatsubtle, due to the nominally global action of complexconjugation. However, for MBL states, which by def-inition permit a tensor product state description, onemay readily construct a well-defined local action of time-reversal . A notable case, is that of spin-1/2 electronswith time-reversal symmetry. In a putative TRS MBLstate of such particles, electronic excitations would ex-hibit a local two-fold Kramers degeneracy, spoiling thestability of the localized phase. In particular, this rulesout the possibility of 2D and 3D TRS electron topologicalinsulators in MBL systems.In fact, this and related obstructions rule out the possi-bility of physically realizing any fermionic topological in-sulator in physically accessible dimensions ( d ≤
3) in the10-fold way classification , for the following reasons.First, any of the topological superconducting classes re-quire a pair condensate, which in ultra-cold atomic sys-tems in which MBL may be realized , implies the ex-istence of a superfluid Goldstone mode which will lead tothermalization . Next, any non-superconductingTI class has either Kramers doublet fermions ( T = − , or chiral edge states – any of which pre-vent symmetry-preserving MBL. Whether any fermionSPT outside the 10-fold way is suitable for MBL protec-tion can be examined on a case-by-case basis using theabove criteria. B. Localization of Anyons
Symmetry can also lead to new topologically or-dered phases and our results immediately imply thatsuch symmetry enriched topological (SET) phases withnon-Abelian symmetry cannot be many-body localized.Moreover, our arguments also rule out MBL protec-tion of classes of SET order in which the global sym-metry group is Abelian, but where the local symmetryaction on fractionalized anyons is projective (requiringmultidimensional local degeneracy) and hence acts likea non-Abelian symmetry . Examples of this classof phases include discrete gauge theories in which theelectric domain walls are decorated by one-dimensionalSPTs, such that the electric charge excitations trans-form as the ends of 1D SPTs, and hence have symmetryprotected degeneracy that prevents symmetry preservingMBL of generic excited states in which such excitationsare present at finite density at random locations.Even without any additional global symmetry, our ar-gument can be naturally generalized to topologically or-dered systems in 2+1 dimensions with non-Abelian any-onic excitations. If such systems could be many-body lo-calized, the finite density of exponentially localized non-Abelian anyons in generic eigenstates would lead to anexponential degeneracy of eigenstates (the quantum di-mension of the anyons playing the role of the dimensionof the irreps in our previous discussion). This forbidsarea-law entangled MBL phases with non-Abelian topo-logical order, and simply reflects the fact that the topo-logical Hilbert space of non-Abelian anyons does not havea local tensor product structure and that the notion oftopological charge cannot be made local. In general, weexpect interacting anyons to either thermalize or to formmore exotic non-ergodic states that cannot be describedin terms of independent l-bits, such as the QCG phase in1D . C. Localizability of anyonic edge modes
The constraints on the localization of anyons dis-cussed above also have consequences for one-dimensional“trenches” of non-Abelian anyons ψ with quantum di-mension d ψ , such as the 1D chain of Majorana boundstates ( d ψ = √
2) that emerges from gapping out theedge of a 2D TI or fractional TI by proximity to alter-nating ferromagnetic and superconducting regions .Focusing on the topological low-energy (in-gap) sector,we can ask whether the 1D topological phase with any-onic edge modes obtained by dimerizing the couplings canbe protected to finite energy density (within the topolog-ical sector) using MBL. In the perfectly dimerized limit,the eigenstates of such a system consist of two dangling anyonic edge modes and of anyonic excitations result-ing from the fusion ψ × ψ on the bulk dimerized bonds.Our discussion implies that an MBL phase away fromthe perfectly dimerized limit can exist if and only if theanyons appearing when fusing ψ with itself all have di-mension one, which can occur only if d ψ = p is an inte-ger — corresponding to the so-called parafermionic zeromodes that generalize Majorana fermions ( p = 2).Only such parafermionic edge modes can be protectedby MBL, which while interesting for topological quan-tum computing applications , are not enough to realizea set of universal quantum gates. One-dimensional chainsof anyons whose braiding would provide a universalgate set (such as Fibonacci anyons for example) cannotbe many-body localized even by strongly dimerizing thecouplings, and instead generically thermalize (at weakdisorder), or form a non-ergodic QCG phase (at strongdisorder) consistent with recent real-space renormaliza-tion group results . V. DISCUSSION AND GENERALIZATIONS
In this paper, we showed that MBL is not possible withsymmetry groups that protect degeneracies ( i.e. thathave multidimensional irreps). This “no-go theorem” re-lies on a specific definition of MBL in terms of local in-tegrability , and the existence of a complete set of lo-cal conserved quantities (“l-bits”) so that all eigenstatesare smoothly connected by a quasilocal unitary trans-formation to a zero correlation length limit. This l-bitpicture has become central to our current understandingof area-law entangled MBL phases, and underlies all ofthe phenomenology of MBL systems (absence of trans-port, logarithmic dephasing etc. ). The existence of MBLphases beyond the l-bit picture is controversial, and couldinclude systems with many-body mobility edges for in-stance . We emphasize here that our argument can benaturally extended beyond the local integrability picture,and would rule out tentative MBL phases that would es-cape the l-bit description. The key point is that our argu-ment relies only on the existence of local excitations overa symmetric eigenstate (say, the groundstate) that trans-form nicely under the symmetry. Even if the l-bit picturebreaks down, excitations of an MBL system should be lo-cal, and we expect that they should naturally transformunder irreps of the symmetry group. A finite density ofsuch local excitations transforming according to irreps ofdimension larger than one would immediately lead to ex-ponentially degenerate eigenstates, which are inherentlyunstable. While this argument can be made essentiallyrigorous within the l-bit picture, we expect the main ideato be fairly general so that it would rule out MBL phaseswithout an l-bit description as well (if such phases doexist).
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In this section, we explicitly construct an example ofa finite depth unitary transformation that converts the SPT to a trivial paramagnet. We consider a discreteversion of the Haldane spin chain with symmetry group Z × Z . The chain contains spin-1/2 degrees of freedom,and has two-sublattices (even and odd sites). The zero-correlation length paramagnet reads: H PM = − L X i =1 h i σ xi . (A1)The symmetry generators are g e = Q i σ x i , g o = Q i σ x i +1 , which flip the spins about the z-axis on theeven and odd sublattice respectively.The SPT phase is described by zero-correlation lengthHamiltonian: H SPT = − L − X i =2 J i σ zi − σ xi σ zi +1 . (A2)Since H SPT is a non-trivial SPT phase, then by definition,we cannot continuously deform the eigenstates of H SPT to those of the trivial paramagnet H PM . Namely, thereis no continuous family of symmetry-preserving unitaryoperators U ( λ ) for λ ∈ [0 , U (0) = 1 and U ( λ =1) = U SPT , where U SPT maps the SPT states to trivialones.However, if we sacrifice the continuity, we can writedown the end result, U ( λ = 1), as a finite-depth unitarycircuit, which is all that is required to establish the lo-cal representation of symmetry. An explicit constructionthat does the job is U SPT = e − iπ/ P i σ zi σ zi +1 , (A3)which takes σ xi σ zi − σ xi σ zi +1 everywhere in the bulkof the chain (see eg. U SPT is finite depth, and preserves the form of the symmetrygenerators g e and g o , except at the boundaries of thesystem.While we have constructed an explicit example of thedesired finite-depth unitary U SPT for this particular sym-metry class, similar constructions can be made for anygeneral SPT class46